Investigating Second Eigenvalue

Investigating Second Eigenvalue

  if (require("PageRank")) {
      library(PageRank)
    }else{
      devtools::install_github("ryangreenup/PageRank")
      library(PageRank)
    }

  library(pacman)
  pacman::p_load(PageRank, devtools, Matrix, igraph, mise, tidyverse, rgl, latex2exp)
#  mise()

Looking at Density

Constants

Define some constants

n <- 20
p <- 1:n/n
beta <- 1:n/n
beta <- runif(n)*100
#sz <- 1:n/n+10
sz <- (1:n/n)*100+10
input_var <- expand.grid("n" = n, "p" = p, "beta" = beta, "size" = sz)
input_var

Function to Build Graph

random_graph <- function(n, p, beta, size) {
      g1 <- igraph::erdos.renyi.game(n = sz, p)
      A <- igraph::get.adjacency(g1) # Row to column
      A <- Matrix::t(A)

      A_dens <- mean(A)
      T      <- PageRank::power_walk_prob_trans(A)
      tr     <- sum(diag(T))
      e2     <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
      A_det  <- det(T)
      return(c(abs(e2), A_dens, A_det, tr))
}

Return results

Map the function

nc <- length(random_graph(1, 1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$n, X$p, X$beta, X$size)
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)

[1] 8000

nrow(Y)

[1] 8000

Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "density", "determinant", "trace")
(data <- cbind(input_var, Y)) %>% head()

data <- data[data$density!=0,]

Plot Results

mod <- lm(eigenvalue2 ~ poly(density, 1), data = data)
mod$residuals %>%  hist(breaks = 90)

hist(rnorm(3000), breaks = 100)

plot(mod)

NA

This is interesting, lets look at a snapshot of it:

pairs(data)

cor(data)

the standard deviation is zero

             n             p
n            1            NA
p           NA  1.0000000000
beta        NA -0.0003878828
size        NA  0.0002679644
eigenvalue2 NA -0.9212105217
density     NA  0.9904278229
determinant NA -0.0165548871
trace       NA -0.8625597471
                     beta          size
n                      NA            NA
p           -0.0003878828  0.0002679644
beta         1.0000000000  0.0003062037
size         0.0003062037  1.0000000000
eigenvalue2  0.0019516406 -0.0001260856
density      0.0007147401  0.0013455370
determinant  0.0014073128  0.0031502969
trace       -0.0017016803 -0.0040214793
              eigenvalue2       density
n                      NA            NA
p           -0.9212105217  0.9904278229
beta         0.0019516406  0.0007147401
size        -0.0001260856  0.0013455370
eigenvalue2  1.0000000000 -0.9287234046
density     -0.9287234046  1.0000000000
determinant  0.0211222176 -0.0164753631
trace        0.6696925620 -0.8728746605
             determinant        trace
n                     NA           NA
p           -0.016554887 -0.862559747
beta         0.001407313 -0.001701680
size         0.003150297 -0.004021479
eigenvalue2  0.021122218  0.669692562
density     -0.016475363 -0.872874661
determinant  1.000000000 -0.005188244
trace       -0.005188244  1.000000000

library(corrplot)
corrplot(cor(data), method = "ellipse", type = "lower")

the standard deviation is zero

names(data)

[1] "n"           "p"          
[3] "beta"        "size"       
[5] "eigenvalue2" "density"    
[7] "determinant" "trace"      

library(scatterplot3d)
shapes = c(16, 17, 18) 
shapes <- shapes[as.numeric(iris$Species)]
scatterplot3d(iris[,1:3], pch = shapes)

d <- data[,names(data) %in% c("eigenvalue2", "density", "determinant")]
scatterplot3d(d, angle = 10)

library(plotly)

mtcars$am[which(mtcars$am == 0)] <- 'Automatic'
mtcars$am[which(mtcars$am == 1)] <- 'Manual'
mtcars$am <- as.factor(mtcars$am)

d <- data[sample(1:nrow(data), 1000),]

fig <- plot_ly(d, x = ~determinant, y = ~density, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'Weight'),
                     yaxis = list(title = 'Gross horsepower'),
                     zaxis = list(title = '1/4 mile time')))

fig

NA

ggplot(data) +
  geom_point(mapping = aes(x = density, y = eigenvalue2, size = beta, color = size )) +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Density of Adjacency Matrix", y = TeX("$\\xi_2$ of $T_{PW}$")) +
  guides(size = FALSE, col = FALSE)

well the spread is heteroskedastic so let’s log transform it:

ggplot(data, aes(x = density, y = log(eigenvalue2))) +
  geom_point(mapping = aes(size = size, color = p, shape = factor(n))) +
#  stat_smooth() +
  scale_size_continuous(range = c(0.1,1.5)) +
  labs(x = "Density of Adjacency Matrix", y = "Second Eigenvalue of Power Walk Transition Probability Matrix")

We could probably model this with a quadratic:

mod_x1 <- lm(log(eigenvalue2) ~ poly(density, 1), data = data)
data$x1 <- predict(mod_x1)

mod_x2 <- lm(log(eigenvalue2) ~ poly(density, 2), data = data)
data$x2 <- predict(mod_x2)

mod_x4 <- lm(log(eigenvalue2) ~ poly(density, 4), data = data)
data$x4 <- predict(mod_x4)

mod_xl <- lm(log(eigenvalue2) ~ log(1-density), data = data)
data$xl <- predict(mod_xl)

mod_df <- data
mod_df_long <- pivot_longer(mod_df, cols = c(x1, x2, x4, xl), names_to = "Model_Type", values_to = "eigenvalue2_mod")
mod_df_long$eigenvalue2_log <- log(mod_df_long$eigenvalue2)


print(c("MSE Linear"  = mean(mod_x1$residuals^2),
        "MSE Quadratic" = mean(mod_x2$residuals^2),
        "MSE Quartic" = mean(mod_x4$residuals^2),
        "MSE Logarithmic" = mean(mod_xl$residuals^2)
        ), 2)

     MSE Linear   MSE Quadratic     MSE Quartic MSE Logarithmic 
          0.078           0.031           0.013           0.023 

ggplot(mod_df_long, aes(x = density)) +
  geom_point(aes(y = eigenvalue2_log), fill = "lightblue", col = "black", size = 0.1, alpha = 0.2) +
  geom_smooth(aes(y = eigenvalue2_mod, col = Model_Type), size = 0.9) +
  labs(col = c("Model \nType")) +
  scale_color_discrete(labels = c("Linear", "Quadratic", "Quartic", "Logarithmic")) +
  labs(x = "Density of A", y = TeX("$\\log \\left( \\xi_2 \\right$ of $\\mathbf{T}$")) +
  theme_linedraw()

data.frame("Model" = c("Linear", "Logarithmic"))
ggplot()

# TODO Change the colour of each model by using pivot_longer
# TODO Make a plot of degree vs RSS, comment that the lack of elbo is evidence to regect
# TODO Try a negative log, any luck? with that
# TODO Does this vary by beta?
# TODO Write it up in a report
# DONE What about a sqrt transform and then a linear model?
  # Leaves too much variance
ggplot(data, aes(x = density, y = log(eigenvalue2))) +
  geom_point(mapping = aes(size = size, alpha = 0.01,  color = size, shape = factor(n))) +
#  stat_smooth() +
  scale_size_continuous(range = c(0.1,1.5)) +
  labs(x = "Density of Adjacency Matrix [ mean(A) ]", y = TeX("$\\log\\left( \\xi_2 \\right)$ of T")) +
  geom_line(aes(x = density, y = x1, lwd = 2)) +
  geom_line(aes(x = density, y = xl, lwd = 3)) 

#  geom_line(aes(x = density, y = x3, lwd = 0.5)) +
#  geom_line(aes(x = density, y = x4, lwd = 0.5)) +
#  geom_line(aes(x = density, y = x5, lwd = 0.5)) +
# geom_line(aes(x = density, y = x6, lwd = 36)) 

Look at the Trace of T

ggplot(data, aes(x = trace , y = log(eigenvalue2))) +
  geom_point(mapping = aes(size = size, color = p, shape = factor(n))) +
#  stat_smooth() +
  scale_size_continuous(range = c(0.1,1.5)) +
  labs(x = "Trace of Transition Matrix", y = TeX("$\\log\\left( \\xi_2 \\right)$ of \\mathbf{W}"))

mod_df <- data

mod_hyp <- lm(log(eigenvalue2) ~ I(trace^(-1)), data = data)
mod_df$hyp <- predict(mod_hyp)

mod_log <- lm(log(eigenvalue2) ~ log(trace), data = data)
mod_df$log <- predict(mod_log)

ggplot(mod_df, aes(x = trace, y = log(eigenvalue2))) +
  geom_point(mapping = aes(size = size, alpha = 0.01,  color = size, shape = factor(n))) +
#  stat_smooth() +
  scale_size_continuous(range = c(0.1,1.5)) +
  labs(x = "Trace of A", y = TeX("$\\log\\left( \\xi_2 \\right)$ of T")) +
  geom_line(aes(x = trace, y = hyp, lwd = 2)) +
  geom_line(aes(x = trace, y = log, lwd = 2))

mod_df <- data

mod_hyp <- lm(log(eigenvalue2) ~ 0 + I(trace^(-1)), data = data)
mod_df$hyp <- predict(mod_hyp)

mod_log <- lm(log(eigenvalue2) ~ 0 + log(trace), data = data)
mod_df$log <- predict(mod_log)


mod_df_long <- pivot_longer(mod_df, cols = c(hyp, log), names_to = "Model_Type", values_to = "eigenvalue2_mod")
mod_df_long$eigenvalue2_log <- log(mod_df_long$eigenvalue2)


print(c("MSE Hyperbolic"  = mean(mod_hyp$residuals^2),
        "MSE Logarithmic" = mean(mod_log$residuals^2)), 2)

ggplot(mod_df_long, aes(x = trace)) +
  geom_point(shape = 23, aes(y = eigenvalue2_log), fill = "lightblue", col = "black", size = 0.7, alpha = 0.4) +
  geom_line(aes(y = eigenvalue2_mod, col = Model_Type), size = 1) +
  labs(col = c("Model \nType")) +
  scale_color_manual(labels = c("Hyperbolic", "Logarithmic"),
                     values = c("indianred", "royalblue")) +
  labs(x = "Trace of A", y = TeX("$\\log \\left( \\xi_2 \\right$ of $\\mathbf{T}$"))

  theme_linedraw()

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 $ plot.caption.position     : chr "panel"
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Model Chi Distribution

NOPE

chival <- dchisq(seq(from = 0, to = 40, length.out = 100), df = 10)*7
index  <- seq(from = 0, to = 2.2, length.out = 100)
chidata  <- data.frame(index = index, chi = chival)
ggplot(data) +
  geom_point(mapping = aes(x = density, y = eigenvalue2, size = beta, color = size, shape = factor(n))) +
  geom_line(data = chidata, mapping = aes(x = index, y = chi)) +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Density of Adjacency Matrix", y = "Second Eigenvalue of Power Walk Transition Probability Matrix")

Looking at Determinant

constants:

n <- 20
p <- 1:n/n
beta <- 1:n/n
beta <- runif(n)*100
sz <- ((1:n)/n)*100+10
input_var <- expand.grid("n" = n, "p" = p, "beta" = beta, "size" = sz)

functions:

random_graph <- function(n, p, beta, size) {
      g1 <- igraph::erdos.renyi.game(n = sz, p)
      A <- igraph::get.adjacency(g1) # Row to column
      A <- Matrix::t(A)

      A_dens <- mean(A)
      T      <- PageRank::power_walk_prob_trans(A)
      e2     <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
      A_det  <- det(A)
      return(c(abs(abs(e2)-0.4), abs(A_det), A_dens))
}

Map the function

nc <- length(random_graph(1, 1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$n, X$p, X$beta, X$size)
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "determinant")
data <- cbind(input_var, Y)

ggplot(data) +
  geom_point(mapping = aes(x = determinant, y = eigenvalue2, size = size, color = beta, shape = factor(n))) +
  scale_size_continuous(range = c(0.1,1)) +
  labs(y = "||e2|-0.4|", x = TeX("$\\left\\lvert A \\right\\rvert $"))

g1 <- igraph::erdos.renyi.game(n = sz, p)
coords <- layout_with_fr(g1, dim = 3)
# plot(g1)
rglplot(g1, layout=coords, size = 0.1)

## Not run: 
g <- make_lattice( c(5,5,5) )
coords <- layout_with_fr(g, dim=3)
rglplot(g, layout=coords)

## End(Not run)

n <- sz <- size <-  10^3
p <- 0.
g1 <- igraph::erdos.renyi.game(n = sz, p)
A <- igraph::get.adjacency(g1) # Row to column
A <- Matrix::t(A)
det(A)

ggplot(data) +
  geom_point(mapping = aes(x = size, y = determinant, size = size, color = beta, shape = factor(n))) +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "size", y = "determinant")

Get Different Data

constants:

n <- 10
p <- 1:n/n
beta <- 1:n/n
beta <- runif(n)*100
sz <- 1:n/n+100
input_var <- expand.grid("n" = n, "p" = p, "beta" = beta, "size" = sz)

functions:

random_graph <- function(n, p, beta, size) {
      g1 <- igraph::erdos.renyi.game(n = sz, p)
      A <- igraph::get.adjacency(g1) # Row to column
      A <- Matrix::t(A)

      A_dens <- mean(A)
      T      <- PageRank::power_walk_prob_trans(A)
      e2     <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
      A_det  <- det(A)
      return(c(abs(e2), A_det, A_dens))
}

Map the function

nc <- length(random_graph(1, 1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$n, X$p, X$beta, X$size)
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "determinant")
data <- cbind(input_var, Y)

chival <- dchisq(seq(from = 0, to = 40, length.out = 100), df = 10)*7
index  <- seq(from = 0, to = 2.2, length.out = 100)
chidata  <- data.frame(index = index, chi = chival)
ggplot(data) +
  geom_point(mapping = aes(x = determinant, y = eigenvalue2, size = size, color = beta, shape = factor(n))) +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Density of Adjacency Matrix", y = "Second Eigenvalue of Power Walk Transition Probability Matrix")

g1 <- igraph::erdos.renyi.game(n = sz, p)
coords <- layout_with_fr(g1, dim = 3)
# plot(g1)
# rglplot(g1, layout=coords, size = 0.1)

## Not run: 
g <- make_lattice( c(5,5,5) )
coords <- layout_with_fr(g, dim=3)
rglplot(g, layout=coords)

## End(Not run)

n <- sz <- size <-  100
p <- 0.4
g1 <- igraph::erdos.renyi.game(n = sz, p)
A <- igraph::get.adjacency(g1) # Row to column
A <- Matrix::t(A)
det(A)